Writing Up Mathematical Solutions

Math homework in university is a lot different than it was in high school. For one thing, there is a lot more emphasis on writing proofs, and a lot less on just performing computations. Many students have difficulty with this, for two reasons: they have difficulty thinking of proofs, and they have difficulty explaining the proofs they have thought of.

There are many tricks you can use to make your life easier in a university mathematics course. There are many simple conventions you can adopt, which will both make it easier for you to express your ideas, and easier for other people to understand them. Making your mathematical ideas easy for other people to understand is important for several reasons:

1. You yourself are not going to remember what you were thinking, six months from now, when you are reviewing your assignments to study for your final exam.
2. It is important that the marker understands the argument you are presenting. If he/she doesn't understand your proof, they might just not believe it, and then you will lose marks needlessly.
3. If you plan to use mathematics for any reason at any point in your future life, it is not only important to learn how to use mathematical tools; it is equally important to learn the skill of communicating mathematical ideas to other people.

As a result, we highly advise you to follow the guidelines to handing in assignments. Not only does it apply to this course, it will benefit you no matter what course in mathematics you will take.

• Clearly state the question and what you need to show. You want to write out the question and the solution as if the marker is reading it for the very first time. In fact, failing to clearly state the question is one of the most common reasons students needlessly lose marks. By restating the problem, you demonstrate whether you understand the question, which is essential to obtaining full marks. If you don't feel ambitious, just copy the question verbatim.
• The solutions to every question must contain justification. Usually this is in the form of computations, but your solution should describe what exactly you are doing. Even if you are asked to sketch a graph, you must explain how you came up with the graph. Even if a problem does not say "prove" or "justify", you are expected to do so.
• Do not submit the first (or any rough) draft of your assignment. Math assignments involve a very high level of exposition. Rough drafts are not acceptable for English essays; they are certainly not acceptable for math assignments as well. Often, finding a proof or a solution to a problem is a long, difficult process where different things you try do not work. Your assignment should not reflect this; it should contain a clear, concise explanation of the final proof you found.
• Make your assignment easy to read. Make it neat and readable. Write in complete English sentences, not in some weird mathematical shorthand that you have invented. Be clear what questions you are answering, and be clear in your answer. If the marker doesn't understand your answer, he might not believe you've answered the question correctly, and you may lose marks as a result.

  Tip! Read your solution out loud exactly the way you have written it and ask yourself whether another student would have understood the solution.

  Another tip! Look at the examples of your calculus text and observe how the author writes out his/her solutions. Your solutions should follow the same style as the author! For proof questions, observe how the author proves major theorems in the text.

  • Many students get in the habit of trying to compress their work into mathematical symbolic shorthand as much as possible, because they think that this somehow makes it more "mathematical". It doesn't; it just makes it impossible to read. There is nothing wrong with expressing an argument in plain old English; far from it, this often makes it clearer and easier to understand. Believe it or not, the reason all this confusing mathematical terminology and cryptic notation was invented was to make it easier to communicate mathematical ideas clearly. Clear communication of mathematics is always the most important goal. When used properly, in the appropriate situation, this cryptic notation CAN facilitate communication. When used unnecessarily, it often serves only to obfuscate.
  • Do not use red pens, since the markers also use red pens to evaluate assignments.
  • Although typing up a solution guarantees a neatly presented assignment, you pay a hefty price for the time it takes to typeset mathematical formulas. We do not recommend typing up math assignments, simply because the extra time you take to type up the problem set could have been used for studying (and not just in this course!).
  • Do not split each page into columns and submit your solutions in multiple columns. It makes it harder to read.
  • Provide plenty of space on the left and right margins so that the marker can provide comments and suggestions on your work. Don't try to cram all your work into one page.
• Try to hand in your solutions to the problems in the order in which the problems were assigned. The marker will be looking for the answer to a certain question at a specific place in the assignment. If it's somewhere else instead, he/she might not find it, and then you might not get marks.
• When dealing with numerical computations, simplify your answer as much as possible, but express numbers algebraically (i.e. leave your solutions in closed form). Do not write decimal expansions in your final answer. For example, 1/3 is far more precise than 0.333333. The square root of 2, divided by 4, is much more precise than 0.353553390592. In fact, you should avoid using a calculator in the entire course. All the exercises in this course can be done without the use of a calculator. The term tests and the final exam prohibit the use of calculators.
• If you have found two or more solutions to the same problem, submit only one. Whenever in doubt, choose the most elegant, straightforward solution. Should you submit more than one solution, the marker will only consider the first solution you have written, and ignore all the others.
• If you think that a statement trivially implies another, you're probably wrong. Don't skip steps. Textbook authors, lecturers, and even the TAs -- you may have noticed -- throw words like "clearly", "trivial", and "obvious" with gleeful abandon. This does not give you licence to do the same. First of all, something that might seem "clear" might actually be quite tricky. Consider that your reader might not be as inspired and enlightened as you are when you write the proof (and that reader might be you, six months from now), so don't hesitate to explain things in pedantic, intelligence-insulting detail. Remember, these assignments are for your benefit! In sum: use your best judgment to determine what is "obvious."

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In sum, assignment marks are deducted for the following errors:

• Failure to state the assumptions of the question and what you are asked to show, compute, or prove.
• Lack of justification for a computational problem. All questions in mathematics require justification, and this means that all your steps toward your solution must contain explanation in complete English sentences (using mathematical notation whenever appropriate, and with correct grammar and spelling!). If you decide that a sketch or a diagram is necessary, make sure you explain what the diagram shows.
• Incorrect mathematical logic and/or computational errors. For example, the statement that "x² > 0 implies x > 0" is an incorrect statement.
• Use of irrelevant mathematical statements, whether they are true or false.
• Lack of clarity and neatness of solutions.

Notice that "having the wrong answer" is not in the list above. In a computational problem, the final answer is always the least important thing a marker will look at.

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